This info is all in your local git history and requires two steps (which can be combined)
What is the last common commit
The commit used to create the PR branch is necessary to know what to compare to, to find this use git merge-base. This is most relevant if the branch has existed for some time, other changes have been merged to master, and the PR branch is out of date with the current master.
$ git merge-base origin/pr-branch-name origin/master
somehash
This is the first commit which is common to both branches, and the commit to compare to.
What are the files that differ between two commits
To get a list of files that differ, use git-diff with the option --name-only
$ git diff --name-only origin/pr-branch-name somehash
src/x
src/y
...
As one command
As a single command that would be:
$ git fetch # if desired
$ PR=pr-branch-name
$ git diff --name-only origin/$PR `git merge-base origin/$PR origin/master`
src/x
src/y
...